modelling and reasoning with vague concepts

VAGUE CONCEPTS AND FUZZY SETS 2.1 Fuzzy Set Theory 2.2 Functionality and Truth-Functionality 2.3 Operational Semantics for Membership Functions 2.3.1 Prototype Semantics 2.3.2 RiskJBetti

Modelling and Reasoning with Vague Concepts

Studies in Computational Intelligence, Volume 12

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word 'meaning' it can be defined thus: the meaning of a word is its use

in language - Ludwig Wittgenstein

2 VAGUE CONCEPTS AND FUZZY SETS

2.1 Fuzzy Set Theory

2.2 Functionality and Truth-Functionality

2.3 Operational Semantics for Membership Functions

2.3.1 Prototype Semantics

2.3.2 RiskJBetting Semantics

2.3.3 Probabilistic Semantics

2.3.3.1 Random Set Semantics

2.3.3.2 Voting and Context Model Semantics

2.3.3.3 Likelihood Semantics

3 LABEL SEMANTICS

3.1 Introduction and Motivation

3.2 Appropriateness Measures and Mass Assignments on Labels 3.3 Label Expressions and A-Sets

3.4 A Voting Model for Label Semantics

3.5 Properties of Appropriateness Measures

3.6 Functional Label Semantics

3.7 Relating Appropriateness Measures

to Dempster-Shafer Theory

xi xix xxi xxiii

vm MODELLING AND REASONING WITH VAGUE CONCEPTS

3.8 Mass Selection Functions based on t-norms

3.9 Alternative Mass Selection Functions

3.10 An Axiomatic Approach to Appropriateness Measures

3.11 Label Semantics as a Model of Assertions

3.12 Relating Label Semantics to Existing Theories of Vagueness

4 MULTI-DIMENSIONAL AND MULTI-INSTANCE LABEL

SEMANTICS

4.1 Descriptions Based on Many Attributes

4.2 Multi-dimensional Label Expressions and A-Sets

4.3 Properties of Multi-dimensional Appropriateness Measures

4.4 Describing Multiple Objects

5 INFORMATION FROM VAGUE CONCEPTS

5.1 Possibility Theory

5.1.1 An Imprecise Probability Interpretation of Possibility

Theory 5.2 The Probability of Fuzzy Sets

5.3 Bayesian Conditioning in Label Semantics

5.4 Possibilistic Conditioning in Label Semantics

5.5 Matching Concepts

5.5.1 Conditional Probability and Possibility given Fuzzy

Sets 5.5.2 Conditional Probability in Label Semantics

5.6 Conditioning From Mass Assignments in Label Semantics

6 LEARNING LINGUISTIC MODELS FROM DATA

Defining Labels for Data Modelling

Bayesian Classification using Mass Relations

6.2.1 Grouping Algorithms for Learning Dependencies in

Mass Relations 6.2.2 Mass Relations based on Clustering Algorithms

Prediction using Mass Relations

Qualitative Information from Mass Relations

Learning Linguistic Decision Trees

6.5.1 The LID3 Algorithm

6.5.2 Forward Merging of Branches

Contents ix 6.7 Query evaluation and Inference from Linguistic Decision Trees 183

7 FUSING KNOWLEDGE AND DATA

7.1 From Label Expressions to Informative Priors

7.2 Combining Label Expressions with Data

7.2.1 Fusion in Classification Problems

7.2.2 Reliability Analysis

8 NON-ADDITIVE APPROPRIATENESS MEASURES

8.1 Properties of Generalised Appropriateness Measures

8.2 Possibilistic Appropriateness Measures

8.3 An Axiomatic Approach to

Generalised Appropriateness Measures

8.4 The Law of Excluded Middle

References

Index

Plot of a possible f, function and its associated k value 14

Diagram showing how fuzzy valuation F, varies with

Diagram showing the rule for evaluating the fuzzy val-

uation of a conjunction at varying levels of scepticism 3 1 Diagram showing the rule for evaluating the fuzzy val-

uation of a negation at varying levels of scepticism 3 1 Diagram showing how the range of scepticism values

for which an individual is considered tall increases with height 33 Diagram showing how the extension of tall varies with

A Functional Calculus for Appropriateness Measures 50 Appropriateness measures for, from left to right, small,

Mass assignments for varying x under the consonant

msf; shown from left to right, m,({small)),

m, ({small, medium)), m, ({medium)),

m, ({medium, large)) and m, ({large)); m, (0) is

equal to m, ({small, medium)) for x E [2,4], is

equal to m,({medium, large)) for x E [6,8] and is

Appropriateness Measure pmedium,,71arge ( x ) under the

consonant msf (solid line) and min(pmedium(x), 1 -

plarge ( x ) ) = pmedium ( x ) (dashed line) corresponding

to pmediUm/\,large ( 2 ) in truth-functional fuzzy logic 56

MODELLING AND REASONING WITH VAGUE CONCEPTS

Mass assignments for varying x under the independent

msf; shown from left to right, m,({small)),

m, ({small, medium)), m, ({medium)),

m,({medium, large)) and m, ({large)); mx(0) is

equal to m, ({small, medium)) for x E [2,4], is

equal to m,({medium, large)) for x E [6,8] and is

zero otherwise

Appropriateness Measure p m e d i z l m ~ ~ ~ a r g e ( x ) under the

independent msf

Plot of values of mx(0) where s = 0.5 p ~ , ( x ) =

p ~ ~ ( x ) = ~ L ~ ( x ) = y and y varies between 0 and

1

Plot of values of m,(Q)) where s = 40 p ~ , ( x ) =

P L ~ ( x ) = p ~ ~ ( x ) = y and y varies between 0 and

The grey cells are those contained within the A-set

Representation of the multi-dimensional A-set,

A(2) ( [ s A h] V [ I A lw]), showing only the focal cells

Fl x F2 The grey cells are those contained within the

A-set

Plot of the appropriateness degree for mediuml A

4argel + medium2

Plot of the appropriateness degree for (mediuml A

llargel + medium2) A (largel + small2)

Appropriateness measures for labels young, middle

aged and old

Mass assignment values form, generated according to

the consonant msf as x varies from 0 to 80

Histogram of the aggregated mass assignment ~ D B

Appropriateness degrees for small, medium and large

Tableau showing the database DB

Mass assignment translation of DB

Tableau showing possibility and probability distribu-

tions for the Hans egg example

List Figures X l l l

Alternative definitions of the conditional distribution

Plot of the conditional density f ( ~ ~ 1 8 , x l ) 115

Plot of the conditional density f (xalO, 6.5) 115

Possibility distributions where n ( x ) = pma ( x ) (black

line) and n ( x ) defined such that n ( x ) = 0 if pma ( x ) <

0.5 and n ( x ) = 2pma ( x ) - 1 if pma ( x ) 2 0.5 (dashed line) 117

X ( m a , a) = {x E R : p,, (x) 2 a, V p middle aged p, (x) < a) 118

Possibility distribution generated from X ( m a , a ) as

defined in figure 5.6 and assuming that a is uniformly

distributed across [0.5,1] and has zero probability for

values less that 0.5

Conditional density given ~ D (i.e B f ( e l m D B ) ) as-

suming a uniform prior The circles along the horizontal

axis represent the original values for age in D B from

which ~ D is derived B

Mass assignment values for m, as x varies from 0 to 80

after normalization

Appropriateness measures for labels young, middle

aged and old after normalization

Conditional density given ~ D (i.e B f ( e l m D B ) ) as-

suming a uniform prior and after normalization The

circles along the horizontal axis represent the original

values for age in D B from which m D B is derived

Labels generated using the uniform discretization method

Labels generated using the uniform discretization method

Figure of Eight Classification Problem

Table showing the mass relation m (el legal) [87]

Histogram of the mass relation m (mllegal) [87]

Plot showing the density function f ( x , y J m (ellegal))

derived from the legal mass relation [87]

Scatter plot showing the classification accuracy of the

label semantics Bayesian classifier [87]

The search tree for attribute groupings using a breadth

first search guided by the improvement measure (defi-

nition 103) [87]

The search tree for attribute groupings using a depth first

search guided by the improvement measure (definition

103) [87]

MODELLING AND REASONING WITH VAGUE CONCEPTS

Results for the Bristol vision database

Results showing percentage accuracy on UCI databases

Scatter plot showing true positive, false negative and

false positive points for the cluster based mass relation

classifier on the figure of eight database

Table of classification accuracy for training (upper

value) and test (lower value) sets for varying numbers of

clusters The number of clusters for cylinders are listed

horizontally and the clusters of prototypes for rocks are

listed vertically

Discretization of a prediction problem where k = 1,

using focal sets The black dots correspond to data

vectors derived from the function g ( x l ) but involving

some measurement error and other noise

The x3 = sin ( x l x 2 2 ) surface defined by a database

of 529 training points

Mass relational model of 2 3 = sin ( X I x x2) [87]

Comparison between the E-SVR prediction, the Mass

Relation prediction and the actual sunspot values

Scatter plots comparing actual sunspot numbers to the

E-SVR System and the mass relational algorithm for the

59 points in the test data set for the years 1921 to 1979

This tableau shows the conditional mass relation

m (elillegal) for the figure of eight classification prob-

lem The grey cells indicate the focal set pairs necessary

for evaluating the rule vll A (v12 V (m2 A s 2 ) ) -, illegal

Linguistic decision tree involving attributes X I , 2 2 , x3

Label semantics interpretation of the LDT given in fig-

Accuracy of LID3 based on different discretization

methods and three other well-known machine learn-

ing algorithms LID3-U signifies LID3 using uniform

discretization and LID3-NU signifies LID3 using non-

uniform discretization [83]

Summary of t-test comparisons of LID3 based on differ- ent discretization methods with three other well-known machine learning algorithms

An illustration of branch merging in LID3

Comparisons of percentage accuracy Acc and the num- ber of branches (rules) 1 LDT 1 with different merging thresholds Tm across a set of UCI datasets The results for Tm = 0 are obtained with n = 2 labels and results for other Tm values are obtained with the number of labels n listed in the second column of the table [83] The change in accuracy and number of leaves as Tm

varies on the breast-w dataset with n = 2 labels

LDT tree for the iris databases generated using LID3 with merging

LDT tree for the iris databases with sets of focal sets converted to linguistic expressions using the simplified 0-mapping

Plot showing the sunspot predictions for the LDT and SVR together with the actual values

Prediction results in MSE on the sunspot prediction problem Results compare the LDT with varying merg- ing thresholds against an SVR and the mass relations method

Scatter plots comparing actual sunspot numbers to the unmerged LID3 results and the merged LID3 results for the 59 points in the test data set for the years 1921 to 1979 Conditional density function f (xl, x 2 1 KB) given knowledge base KB

Prior mass relation p m on fll x f12

Conditional mass relation ~ K on BR 1 x f12

Gaussian appropriateness measures for small, medium, and large

Mass assignments on 2LA generated by the appropri- ateness measures in figure 7.4 under the consonant msf Densities generated by the N I S K B (dashed line) and the MENCSKB(solid line) methods, assuming a uni- form initial prior

Densities generated from V ( N C K B ) asps ranges from 0.3 to 0.6, assuming a uniform initial prior

xvi

7.8

MODELLING AND REASONING WITH VAGUE CONCEPTS

Nearest consistent density (solid line) and normalised

independent density (dashed line), assuming a uniform

Scatter plot of classification results using independent

mass relations to model each class Crosses represent

points correctly classified and zero represent points in-

Density derived from the independent mass relation for

Scatter plot of classification results using expert knowl-

edge only to model each class Crosses represent points

correctly classified and zero represent points incorrectly

classified

Density generated from KBleSal for legal

Scatter plot of classification results using both back-

ground knowledge and independent mass relations to

model each class Crosses represent points correctly

classified and zero represent points incorrectly classified 208 Density generated from fused model for legal 209 Results for figure of eight classification problem 209 Figure of eight classification results based on uncertain

Scatter plot of classification results using uncertain ex-

pert knowledge only to model each class Crosses repre-

sent points correctly classified and zero represent points

Density generated from uncertain knowledge base for legal 21 1 Scatter plot of classification results using the fused

model from uncertain expert knowledge and data to

model each class Crosses represent points correctly

classified and zeros represent points incorrectly classified 21 1 Density generated from uncertain knowledge base fused

Classification of the parameter space in condition as-

sessment guidance for flood defence revetments [13] 214 Contour plot showing the label based classification of

Contour plot showing the density derived from the in-

dependent mass relation and based on the data only 215

7.24 Contour plot showing the density derived from the inde-

pendent mass relation fused with the fuzzy classification

7.25 Regions partitioning the doubtful area based on label

7.26 Contour plot showing the density derived from the inde-

pendent mass relation fused with uncertain description

Preface

Vague concepts are intrinsic to human communication Somehow it would seems that vagueness is central to the flexibility and robustness of natural lan- guage descriptions If we were to insist on precise concept definitions then

we would be able to assert very little with any degree of confidence In many cases our perceptions simply do not provide sufficient information to allow us

to verify that a set of formal conditions are met Our decision to describe an individual as 'tall' is not generally based on any kind of accurate measurement

of their height Indeed it is part of the power of human concepts that they do not require us to make such fine judgements They are robust to the imprecision of our perceptions, while still allowing us to convey useful, and sometimes vital, information The study of vagueness in Artificial Intelligence (AI) is therefore motivated by the desire to incorporate this robustness and flexibility into intel- ligent computer systems This goal, however, requires a formal model of vague concepts that will allow us to quantify and manipulate the uncertainty resulting from their use as a means of passing information between autonomous agents

I first became interested in these issues while working with Jim Baldwin

to develop a theory of the probability of fuzzy events based on mass assign- ments Fuzzy set theory has been the dominant theory of vagueness in A1 since its introduction by Lotfi Zadeh in 1965 and its subsequent successful appli- cation in the area of automatic control Mass assignment theory provides an attractive model of fuzzy sets, but I became increasingly frustrated with a range

of technical problems and unintuitive properties that seemed inherent to both theories For example, it proved to be very difficult to devise a measure of con- ditional probability for fuzzy sets, that satisfied all of a minimal set of intuitive properties Also, mass assignment theory provides no real justification for the truth-functionality assumption central to fuzzy set theory

This volume is the result of my attempts to understand and resolve some of these fundamental issues and problems, in order to provide a coherent frame- work for modelling and reasoning with vague concepts It is also an attempt to

develop such a framework as can be applied in practical problems concerning automated reasoning, knowledge representation, learning and fusion I do not believe A1 research should be carried out in isolation from potential applica- tions In essence A1 is an applied subject Instead, I am committed to the idea that theoretical development should be informed by complex practical prob- lems, through the direct application of theories as they are developed Hence,

I have dedicated a significant proportion of this book to presenting the appli- cation of the proposed framework in the areas of data analysis, data mining and information fusion, in the hope that this will give the reader at least some indication as to the utility of the more theoretical ideas

Finally, I believe that much of the controversy in the A1 community sur- rounding fuzzy set theory and its application arises from the lack of a clear operational semantics for fuzzy membership functions, consistent with their truth-functional calculus Such an interpretation is important for any theory to ensure that its not based on an ad hoc, if internally consistent, set of inference processes It is also vital in knowledge elicitation, to allow for the translation

of uncertainty judgements into quantitative values For this reason there will be

a semantic focus throughout this volume, with the aim of identifying possible operational interpretations for the uncertainty measures discussed

Acknowledgments

Time is becoming an increasingly rare commodity in this frenetic age Yet time, time to organise one's thoughts and then to commit them to paper, is exactly what is required for writing a book For this reason I would like to begin

by thanking the Department of Engineering Mathematics at the University of Bristol for allowing me a six month sabbatical to work on this project Without the freedom from other academic duties I simply would not have been able to complete this volume

As well as time, any kind of creative endeavour requires a stimulating envi- ronment and I would like to thank my colleagues in Bristol for providing just such an environment I was also very lucky to be able to spend three months during the summer of 2004 visiting the Secci6 Matemhtiques i Informhtica at the Universidad Politkcnica de Cataluiia I would like to thank Jordi Recasens for his kindness during this visit and for many stimulating discussions on the nature of fuzziness and similarity I am also grateful to the Spanish govern- ment for funding my stay at UPC under the scheme 'Ayudas para movilidad de profesores de universidad e investigadores Espaiioles y extranjeros'

Over the last few years I have been very fortunate to have had a number of very talented PhD students working on projects relating to the label semantics framework In particular, I would like to thank Nick Randon and Zengchang Qin who between them have developed and implemented many of the learning algorithms described in the later chapters of this book

Finally a life with only work would be impoverished indeed and I would like to thank my wonderful family for everything else To my mother, my wife Pepa, and daughters Ana and Julia - gracias por su amor y su apoyo

Fuzzy set theory, since its inception in 1965, has aroused many contro- versies, possibly because, for the first time, imprecision, especially linguistic imprecision, was considered as an object of investigation from an engineering point of view Before this date, there had already been proposals and disputes around the issue of vagueness in philosophical circles, but never before had the vague nature of linguistic information been considered as an important issue

in engineering sciences It is to the merit of Lotfi Zadeh that he pushed this issue to the forefront of information engineering, claiming that imprecise ver- bal knowledge, suitably formalized, could be relevant in automating control or problem-solving tasks

Fuzzy sets are simple mathematical tools for modelling linguistic informa- tion Indeed they operate a simple shift from Boolean logic, assuming that there

is more to "truth-values" than being true or being false Intermediate cases, like

"half-true" make sense as well, just like a bottle can be half-full So, a fuzzy set is just a set with blurred boundaries and with a gradual notion of member- ship Moreover, the truth-functionality of Boolean logic was kept, yielding a wealth of formal aggregation functions for the representation of conjunction, disjunction and other connectives This proposal also grounds fuzzy set theory

in the tradition of many-valued logics This approach seems to have generated misunderstandings in view of several critiques faced by the theory of fuzzy sets A basic reason for the reluctance in established scientific circles to accept fuzzy set theory is probably the fact that while this very abstract theory had an immediate intuitive appeal which prompted the development of many practical applications, the notion of membership functions had not yet been equipped with clear operational semantics Namely, it is hard to understand the meaning

of the number 0.7 on the unit interval, in a statement like "Mr Smith is tall to degree 0.7", even if it clearly suggests that this person is not tall to the largest extent

xxiv MODELLING AND REASONING WITH VAGUE CONCEPTS

This lack of operational semantics, and of measurement paradigms for mem- bership degrees was compensated for by ad hoc techniques like triangular fuzzy sets, and fuzzy partitions of the reals, that proved instrumental for addressing practical problems Nevertheless, degrees of membership were confused with degrees of probability, and orthodox probabilists sometimes accused the fuzzy set community of using a mistaken surrogate probability calculus, the main ar- gument being the truth-functionality assumption, which is mathematically in- consistent in probability theory Besides, there are still very few measurement- theoretic works in fuzzy set theory, while this would be a very natural way of addressing the issue of the meaning of membership grades Apparently, most measurement-theory specialists did not bother giving it a try

Important progress in the understanding of membership functions was made

by relating fuzzy sets and random sets: while membership functions are not probability distributions, they can be viewed as one-point coverage functions

of random sets, and, as such, can be seen as upper probability bounds This is the right connection, if any, between fuzzy sets and probability But the price paid is the lack of universal truth-functionality

The elegant and deep monograph written by Jon Lawry adopts this point of view on membership functions, for the purpose of modelling linguistic scales, with timely applications to data-mining and decision-tree learning However it adopts a very original point of view While the traditional random set approach

to fuzzy sets considers realisations as subsets of some numerical reference scale (like a scale of heights for "short and tall"), the author assumes they are subsets

of the set of labels, obtained from answering yestno questions about how to label objects This approach has the merit of not requiring an underlying nu- merical universe for label semantics Another highlight of this book is the lucid discussion concerning the truth-functionality assumption, and the proposal of

a weaker, yet tractable, "functionality" assumption, where independent atomic labels play a major role In this framework, many fuzzy connectives can be given an operational meaning This book offers an unusually coherent and comprehensive, mathematically sound, intuitively plausible, potentially useful, approach to linguistic variables in the scope of knowledge engineering

Of course, one may object to the author's view of linguistic variables The proposed framework is certainly just one among many possible other views

of membership functions Especially, one may argue that founding the mea- surement of gradual entities on yes-no responses to labelling questions may sound like a paradox, and does not properly account for the non-Boolean na- ture of gradual notions The underlying issue is whether fuzzy predicates are fuzzy because their crisp extension is partially unknown, or because they are intrinsically gradual in the mind of individuals (so that there just does not exist such a thing as "the unknown crisp extension of a fuzzy predicate") Although

it sounds like splitting hairs, answering this question one way or another has

drastic impact on the modelling of connectives and the overall structure of the underlying logic For instance if "tall" means a certain interval of heights I can- not precisely describe, then "not tall" just means the complement of this interval

So, even though I cannot precisely spot the boundary of the extension of "tall",

I can claim that being "tall and not tall" is an outright contradiction, and "being tall or not tall" expresses a tautology This view enforces the laws of contra- diction and excluded-middle, thus forbidding truth-functionality of connectives acting on numerical membership functions However, if fuzzy predicates are seen as intrinsically gradual, then "tall" and "not tall" are allowed to overlap, then the underlying structure is no longer Boolean and there is room for truth- functionality Fine, would say the author, but what is the measurement setting that yields such a non-Boolean structure and provides for a clear intuition of membership grades? Such a setting does not exist yet and its discovery remains

as an open challenge

Indeed, while the claim for intrinsically gradual categories is legitimate, most interpretative settings for membership grades proposed so far (random sets, similarity relations, utility ) seem to be at odds with the truth-functionality assumption, although the latter is perfectly self-consistent from a mathematical point of view (despite what some researchers mistakenly claimed in the past)

It is the merit of this book that it addresses the apparent conflict between truth- functionality and operational semantics of fuzzy sets in an upfront way, and that it provides one fully-fledged elegant solution to the debate No doubt this somewhat provocative but scientifically solid book will prompt useful debates

on the nature of fuzziness, and that new alternative proposals will be triggered

by its in-depth study The author must be commended for an extensive work that highlights an important issue in fuzzy set theory, that was perhaps too cautiously neglected by its followers, and too aggressively, sometimes misleadingly, ar- gued about, by its opponents from more established fields

Didier Dubois, Directeur de Recherches IRIT -UPS -CNRS

1 18 Route de Narbonne

3 1062 Toulouse Cedex Toulouse , France

Chapter 1

INTRODUCTION

Every day, in our use of natural language, we make decisions about how to label objects, instances and events, and about what we should assert in order

to best describe them These decisions are based on our partial knowledge

of the labelling conventions employed by the population, within a particular linguistic context Such knowledge is obtained through our experience of the use of language and particularly through the assertions given by others Since these experiences will differ between individuals and since as humans we are not all identical, our knowledge of labelling conventions and our subsequent use of labels will also be different However, in order for us to communicate effectively there must also be very significant similarities in our use of label de- scriptions Indeed we can perhaps view the labelling conventions of a language

as an emergent phenomena resulting from the interaction between similar but subtly different individuals Now given that knowledge of these conventions

is, at best, only partial, resulting as it does from a process of interpolation and extrapolation, we will tend to be uncertain about how to label any particular instance Hence, labelling decisions must then be made in the presence of this uncertainty and based on our limited knowledge of language rules and conven-

tions A consequence of this uncertainty is that individuals will find it difficult

to identify the boundaries amongst instances at which concepts cease to be applicable as valid descriptions

The model of vague concepts presented in this volume is fundamentally linked to the above view of labelling and the uncertainty associated with the decisions that an intelligent agent must make when describing an instance Central to this view is the assumption that agents believe in the meaningfulness

of these decisions In other words, they believe that there is a 'correct way' to use words in order to convey information to other agents who share the same (or similar) linguistic background By way of justification we would argue that

such a stance is natural on the part of an agent who must make crisp decisions about what words and labels to use across a range of contexts This view would seem to be consistent with the epistemic model of vagueness proposed by Williamson [I081 but where the uncertainty about concept definition is identified

as being linguistic in nature On the other hand, there does seem to be a subtle difference between the two positions in that our view does not assume the actual existence of some objectively correct (but unknown) definition of a vague concept Rather individuals assume that there is a fixed (but partially known) labelling convention to which they should adhere if they wish to be understood The actual rules for labelling then emerge from the interaction between individuals making such an assumption Perhaps we might say that agents find it useful to adopt an 'epistemic stance' regarding the applicability

of vague concepts In fact our view seems closer to that of Parikh [77] when

he argues for an 'anti-representational' view of vagueness based on the use of words through language games

We must now pause to clarify that this volume is not intended to be primarily concerned with the philosophy of vagueness Instead it is an attempt to develop

a formal quantitative framework to capture the uncertainty associated with the use of vague concepts, and which can then be applied in artificial intelligence systems However, the underlying philosophy is crucial since in concurrence with Walley [I021 we believe that, to be useful, any formal model of uncertainty must have a clear operational semantics In the current context this means that our model should be based on a clearly defined interpretation of vague con- cepts A formal framework of this kind can then allow for the representation of high-level linguistic information in a range of application areas In this volume, however, we shall attempt to demonstrate the utility of our framework by fo- cussing particularly on the problem of learning from data and from background knowledge

In many emerging information technologies there is a clear need for auto- mated learning from data, usually collected in the form of large databases In

an age where technology allows the storage of large amounts of data, it is nec- essary to find a means of extracting the information contained to provide useful models In machine learning the fundamental goal is to infer robust models with good generalization and predictive accuracy Certainly for some applica- tions this is all that is required However, it is also often required that the learnt models should be relatively easy to interpret One should be able to understand the rules or procedures applied to arrive at a certain prediction or decision This

is particularly important in critical applications where the consequences of a wrong decision are extremely negative Furthermore, it may be that some kind

of qualitative understanding of the system is required rather than simply a 'black box' model that can predict (however accurately) it's behaviour For example, large companies such as supermarkets, high street stores and banks continuously

Introduction 3 collect a stream of data relating to the behaviour of their customers Such data must be analysed in such a way as to give an understanding of important trends and relationships and to provide flexible models that can be used for a range

of decision making tasks From this perspective a representational framework based on first order logic combined with a model of the uncertainty associated with using natural language labels, can provide a very useful tool The high- level logical language means that models can be expressed in terms of rules relating different parameters and attributes Also, the underlying vagueness of the label expressions used allows for more natural descriptions of the systems, for more robust models and for improved generalisation

In many modelling problems there is significant background knowledge available from domain experts If this can be elicited in an appropriate form and then fused with knowledge inferred from data then this can lead to significant improvements in the accuracy of the models obtained For example, taking into account background knowledge regarding attribute dependencies, can of- ten simplify the learning process and allow the use of simpler, more transparent, models However, the process of knowledge elicitation is notoriously difficult, particularly if knowledge must be translated into a form unfamiliar to the ex- pert Alternatively, if the expert is permitted to provide their information as rules of thumb expressed in natural language then this gives added flexibility in the elicitation process By translating such knowledge into a formal framework

we can then investigate problems of inductive reasoning and fusion in a much more conceptually precise way

The increased use of natural language formalisms in computing and scientific modelling is the central goal of Zadeh's 'computing with words' programme [I 171 Zadeh proposes the use of an extended constraint language, referred to

as precisiated natural language [118], and based fundamentally on fuzzy sets Fuzzy set theory and fuzzy logic, first introduced by Zadeh in [I 101, have been the dominant methodology for modelling vagueness in A1 for the past four or five decades, and for which there is now a significant body of research literature investigating both formal properties and a wide range of applications Zadeh's framework introduces the notion of a linguistic variable 11121-[114], defined to

be a variable that takes as values natural language terms such as large, small, medium etc and where the meaning of these words is given by fuzzy sets on some underlying domain of discourse An alternative linguistic framework has been proposed by Schwartz in a series of papers including [92] and [93] This methodology differs from that of Zadeh in that it is based largely on inference rules at the symbolic level rather than on underlying fuzzy sets While the mo- tivation for the framework proposed in this volume is similar to the computing with words paradigm and the work of Schwartz, the underlying calculus and its interpretation are quite different Nonetheless, given the importance and

success of fuzzy set theory, we shall throughout make comparisons between it and our new framework

In chapter 2 we overview the use of fuzzy set theory as a framework for describing vague concepts While providing a brief introduction to the basic mathematics underlying the theory, the main focus of the chapter will be on the interpretation or operational semantics of fuzzy sets rather than on their formal properties This emphasis on semantics is motivated by the conviction that in order to provide an effective model of vagueness or uncertainty, the measures associated with such a framework must have a clearly understood meaning Furthermore, this meaningful should be operational, especially in the sense that it aids the elicitation of knowledge and allows for the expression of clearly interpretable models In particular, we shall discuss the main interpretations

of fuzzy membership functions that have been proposed in the literature and consider whether each is consistent with a truth-functional calculus like that proposed by Zadeh [110] Also, in the light of results by Dubois and Prade [20],

we shall emphasise the strength of the truth-functionality assumption itself and suggest that a weaker form of functionality may be more appropriate Overall

we shall take the view that the concept of fuzzy sets itself has a number of plausible interpretations but none of these provides an acceptable justification for the assumption of truth-functionality

Chapter 3 introduces the label semantics framework for modelling vague concepts in AI This attempts to formalise many of the ideas outlined in this first part of this chapter by focusing on quantifying the uncertainty that an in- telligent agent has about the labelling conventions of the population in which hetshe is a member, and specifically the uncertainty about what labels are ap- propriate to describe any particular given instance This is achieved through the introduction of two strongly related measures of uncertainty, the first quantify- ing the agents belief that a particular expression is appropriate to describe an instance and the second quantifying the agents uncertainty about which amongst the set of basic labels, are appropriate to describe the instance Through the interaction between these two measures it is shown that label semantics can provide a functional but never truth-functional calculi, with which to reason about vague concept labels The functionality of such a calculus can be related

to some of the combination operators for conjunction and disjunction used in fuzzy logic, however, in label semantics such operators can only be applied to simple conjunctions and disjunctions of labels and not to more complex logical expressions Also, in chapter 3 it is shown how this framework can be used to

investigate models of assertion, whereby an agent must choose what particular logical expression to use in order to describe an instance This must take ac- count of the specificity and logical form of the expression as well as its level

of appropriateness as a description Finally, in this chapter we will relate the

Introduction 5 label semantics view of vagueness to a number of other theories of vagueness proposed within the philosophy literature

The version of label semantics outlined in chapter 3 is effectively 1- dimensional in that it is based on the assumption that each object is described

in terms of only one attribute In chapter 4 we extend the theory to allow for multiple attributes and also by allowing descriptions of more than one object Also in this chapter we show how multi-dimensional label expressions (object descriptions referring to more than one attribute) can be used to model input- output relationships in complex systems Finally, by extending our measures

of uncertainty to descriptions of multiple objects we consider how such mea- sures can be aggregated across a database This will have clear applications in chapters 6 and 7 where we apply our framework to data modelling and learning Another fundamental consideration regarding the use of vague concepts con- cerns the information that they provide about the object or objects being de-

scribed If we are told that 'Bill is tall' what information does this give us about

Bill and specifically about Bill's height? In chapter 5 we investigate a number

of different theories of the information provided by vague concepts A number

of these relate to fuzzy set theory and include Zadeh's possibility theory [1 151 and probability theory of fuzzy events [ I l l ] Chapter 5 also presents a number

of alternative models of probability conditional on vague information, based

on both mass assignment theory [5]-[7] and on the label semantics framework introduced in chapters 3 and 4 In addition to the information provided about objects we also consider the information provided by vague concepts about other vague concepts In this context we investigate measures of conditional matching of expressions and in particular conditional probability For the latter

a number of conditional probability measures are proposed based on Zadeh's framework, mass assignment theory and label semantics These are then as- sessed in terms of whether or not they satisfy a number of axioms and properties characterising certain intuitive epistemic principles Finally, in the label seman- tics model we consider the case of conditioning on knowledge taking the form

of a distribution (mass assignment) across sets of possible labels

Chapters 6 and 7 concern the use label semantics as a representation frame- work for data modelling Chapter 6 focuses on the development of learning algorithms for both classification and prediction (sometimes called regression) tasks Both types of problem require the learning of input-output functional re- lationships from data, where for classification problems the outputs are discrete classes and for prediction problems the outputs are real numbers Two types

of linguistic models are introduced, capable of carrying out both types of task Mass relations quantify the link between sets of appropriate labels on inputs and either output classes or appropriate label sets for describing a real valued output Linguistic decision trees are tree structured sets of quantified rules, where for the each rule the antecedent is a conjunction of label expressions

describing input attributes and the consequent is either a class or an expression describing the real-valued output Associated with each such rule is the condi- tional probability of the consequent given the antecedent Consequently rules

of this form can model both concept vagueness and uncertainty inherent in the training data

Chapter 7 investigates the use of label semantics framework for data and knowledge fusion For example, it is shown how the conditional measures introduced in chapter 5 can be used to generate informative prior distributions

on the domain of discourse from background knowledge expressed in terms

of linguistic statements This approach can be extended to the case where the available knowledge is uncertain In such cases a family of priors are identified and additional constraints must be introduced in order to obtain a unique solution In the second part of chapter 7 we focus on the use of mass relations to fuse background knowledge and data This again incorporates conditioning methods described in chapter 5 In this context we investigate fusion in classification problems and also in reliability analysis for engineering systems The latter is particularly relevant since although empirical data is usually limited there is often other qualitative information available Overall

we demonstrate that the incorporation of appropriate background knowledge results in more accurate and informative models

In Chapter 8 we return to consider theoretical issues concerning the label semantics calculus, by introducing non-additive measures of appropriateness These provides a general theory incorporating a wider range of measures and allowing for a more flexible, if somewhat more abstract, calculus This chapter does show that the assumption of additivity is not unavoidable in the devel- opment of the label semantics framework However, the application of these generalised measures in A1 remains to be investigated

Introduction 7

Notes

1 This terminology is inspired by Dennett's idea of an 'intentional stance' in

[la

VAGUE CONCEPTS AND FUZZY SETS

Vague or fuzzy concepts are fundamental to natural language, playing a central role in communications between individuals within a shared linguistic context In fact Russell [90] even goes so far as to claim that all natural language concepts are vague Yet often vague concepts are either viewed as problematic because of their susceptibility to Sorities paradoxes or at least as somehow 'second rate' when compared with the more precise concepts of the physical sciences, mathematics and formal logic This view, however, does not properly take account of the fact that vague concepts seem to be an effective means

of communicating information and meaning Sometimes more effective, in fact, than precise alternatives Somehow, knowing that 'the robber was tall'

is more useful to the police patrolling the streets, searching for suspects, than the more precise knowledge that 'the robber was exactly 1.8 metres in height' But what is the nature of the information conveyed by fuzzy statements such

as 'the robber was tall' and what makes it so useful? It is an attempt to answer this and other related questions that will be the central theme of this volume Throughout, we shall unashamedly adopt an Artificial Intelligence perspective

on vague concepts and not even attempt to resolve longstanding philosophical problems such as Sorities paradoxes Instead, we will focus on developing an understanding of how an intelligent agent can use vague concepts to convey information and meaning as part of a general strategy for practical reasoning and decision making Such an agent could be an artificial intelligence program

or a human, but the implicit assumption is that their use of vague concepts is governed by some underlying internally consistent strategy or algorithm For simplicity this agent will be referred to using the pronoun You This convention

is borrowed from Smets work on the Transferable Belief Model (see for example [97]) although the focus of this work is quite different We shall immediately attempt to reduce the enormity of our task by restricting the type of vague

10 MODELLING AND REASONING WITH VAGUE CONCEPTS concept to be considered For the purposes of this volume we shall restrict our attention to concepts as identified by words such as adjectives or nouns that can

be used to describe a object or instance For such an expression 0 it should be meaningful to state that 'x is 0' or that 'x is a 0' l Given a universe of discourse

R containing a set of objects or instances to be described, it is assumed that all relevant expression can be generated recursively from a finite set of basic labels

LA Operators for combining expressions are restricted to the standard logical connectives of negation ( l ) , conjunction (A), disjunction (V) and implication

(+) Hence, the set of label expressions identifying vague concepts can be formally defined as follows:

DEFINITION 1 Label Expressions

The set of label expressions of LA, LE, is dejined recursively as follows:

(ii) IfO, cp E LE then 10,O A cp, 0 V cp, 0 + cp E LE

For example, R could be the set of suspects for a rob- bery and L A might correspond to a set of basic labels used by police for identifying individuals, such as L A =

{tall, medium, short, medium build, heavy build, stocky, t h i n ,

blue eyes, brown eyes .) In this case possible expressions in LE include

medium A l t a l l A brown eyes ('medium but not tall with brown eyes') and short A (medium build V heavy build) ('short with medium or heavy build')

Since it was first proposed by Zadeh in 1965 [I 101 the treatment of vague concepts in artificial intelligence has been dominated by fuzzy set theory In this volume, we will argue that aspects of this theory are difficult to justify, and propose an alternative perspective on vague concepts This in turn will lead us to develop a new mathematical framework for modelling and reasoning with imprecise concepts We begin, however, in this first chapter by reviewing current theories of vague concepts based on fuzzy set theory This review will take a semantic, rather than purely axiomatic, perspective and investigate

a number of proposed operational interpretations of fuzzy sets, taking into account their consistency with the mathematical calculus of fuzzy theory

The theory of fuzzy sets, based on a truth-functional calculus proposed by Zadeh [I 101, is centred around the extension of classical set theoretic operations such as union and intersection to the non-binary case Fuzzy sets are generali- sations of classical (crisp) sets that allow elements to have partial membership Every crisp set A is characterised by its membership function X A : R t { O , 1 ) where X A ( x ) = 1 if and only if x E A and where X A ( x ) = 0 otherwise For

fuzzy sets this definition is extended so that XA : R + [O, 11 allowing x to have partial membership XA (x) in A

Fuzzy sets can be applied directly to model vague concepts through the notion

of extension The extension of a crisp (non-fuzzy) concept 0 is taken to be those objects in the universe R which satisfy 0 i.e {x E R : 'x is 0' is true) In the case of vague concepts it is simply assumed that some elements have only partial membership in the extension In other words, the extension of a vague concept

is taken to be a fuzzy set Now in order to avoid any cumbersome notation

we shall also use 0 to denote the extension of an expression 0 E LE Hence, according to fuzzy set theory [I 101 the extension of a vague concept 0 is defined

by a fuzzy set membership function xe : R + [0, 11 Now given this possible framework You are immediately faced with a difficult computational problem Even for a finite basic label set LA there are infinitely many expressions in LE generated by the recursive definition 1 You cannot hope to explicitly define a membership function for any but a small subset of these expressions Fuzzy set theory attempts to overcome this problem by providing a mechanism according

to which the value for xe (x) can be determined uniquely from the values

XL (x) : L E LE This is achieved by defining a mapping function for each

of the standard logical connectives; f A : [O, 112 -+ [O,l], f v : [O, 112 + [O, 11, f+ : [O, 112 + [O,1] and f, : [O, 11 -+ [O, I] The value of xe (x) for any expression 0 and value x E R can then be determined from XL (x) : L E LA

according to the following recursive rules:

This assumption is referred to as truth-functionality due to the fact that it extends the recursive mechanism for determining the truth-values of compound sentences from propositional variables in propositional logic In fact, a funda- mental assumption of fuzzy set theory is that the above functions coincide with the classical logic operators in the limit case when xe (x) , x,+, (x) E (0, 1) Beyond this constraint it is somewhat unclear as to what should be the precise definition of these combination functions However, there is a wide consensus that fA, fv and f, [54] should satisfy the following sets of axioms:

Conjunction

C1 'da E [ O , 1 ] f A (a, 1) = a

C2 'da, b, c E [O, 11 if b 5 c then f A (a, b) 5 f A (a, c)

12 MODELLING AND REASONING WITH VAGUE CONCEPTS

D l 'da E [O,1] f v ( a , 0 ) = a

D2 'da, b, c E [0, 11 if b 5 c then fv ( a , b) < fv ( a , c )

Negation

N1 f, ( 1 ) = 0 and f, ( 0 ) = 1

N2 f, is a continuous function on [O,1]

N3 f, is a decreasing function on [O,1]

N4 'da E [ O , l ] f , ( f , ( a ) ) = a

Axioms C1-C4 mean that f A is a triangular norm or (t-norm) as defined by [94]

in the context of probabilistic metric spaces Similarly according to D1-D4 fv

is a triangular conorm (t-conorm) An infinite family of functions satisfy the t-norm and t-conorm axioms including fA = min and fv = max proposed by Zadeh [I 101 Other possibilities are, for conjunction, fA ( a , b) = a x b and

f A ( a , b) = max ( 0 , a + b - 1 ) and, for disjunction, fv ( a , b) = a + b - a x b

and min ( 1 , a + b) Indeed it can be shown [54] that fA and fv are bounded as follows:

where fA - is the drastic t-norm defined by:

a : b = l ' d a , b ~ [ O , l ] & ( a , b ) =

0 : otherwise

and fv is the drastic t-conorm defined by:

a : b=O 'da, b E [O,1] fv ( a , b) = b : a = O

1 : otherwise

Interestingly, adding the additional idempotence axioms restricts t-norms to min and t-conorms to max:

For a , b E [O,1] suppose a I b then

by axioms CI, C2 and C5 and therefore f A ( a , b) = a = min ( a , b)

Alternatively, for a , b E [O,1] suppose b 5 a then

by axioms CI, C2, C3 and C.5 and therefore f A ( a , b) = b = min ( a , b)

THEOREM 3 fv satisfies Dl-D5 ifand only i f f v = max

The most common negation function f, proposed is f, ( a ) = 1 - a although

there are again infinitely many possibilities including, for example, the family

of parameterised functions defined by:

Somewhat surprisingly, however, all negation functions essentially turn out to

be rescalings of f, ( a ) = 1 - a as can be seen from the following theorem due

14 MODELLING AND REASONING WITH VAGUE CONCEPTS

Then g ( 0 ) = 0, g ( 1 ) = 1 and g ( k ) = 0.5 Also, it is easy to check that g is strictly increasing and continuous and hence onto

Finally, for x 5 k f , ( x ) 2 f , ( I c ) = k (by N3) and therefore

Similarly, for x > k then f , ( x ) 5 f , ( k ) = k and therefore

Figure 2.1: Plot of a possible f, function and its associated k value

In view of this rather strong result we shall now assume that f , ( a ) = 1 - a

and move on to consider possible relationships between t-norms and t-conorms Most of the constraints relating t-norms and t-conorms come from the impo- sition of classical logic equivalences on vague concepts Typical of these is the duality relationship that emerges from the assumption that vague concepts satisfy de Morgan's Law i.e that 8 V cp r 1 0 A l c p In the context of truth- functional fuzzy set theory this means that:

t-norm fA ( a , b) t-conorm dual fv (a, b)

min ( a , b) max ( a , b)

a + b - a x b

m a x ( O , a + b - 1 ) min ( 1 , a + b)

Figure 2.2: t-norms with associated dual t-conorms

In additional, to constraints based on classical logical equivalences it might also be desirable for fuzzy memberships to be additive in the sense that

This generates the following equation relating t-norms and t-conorms:

As described in the previous section fuzzy logic [I 101 is truth-functional,

a property which significantly reduces both the complexity and storage re- quirements of the calculus Truth-functionality is, however, a rather strong assumption that significantly reduces the number of standard Boolean prop- erties that can be satisfied by the calculus For instance, Dubois and Prade

16 MODELLING AND REASONING WITH VAGUE CONCEPTS

[20] effectively showed that no non-trivial truth-functional calculus can satisfy idempotence together with the law of excluded middle ,

I f x is truth-functional and satisjies both idempotence and the law of excluded middle then VO E LE Vx E R xe ( x ) E { O , l )

Proof

From theorem 3 we have that the only idempotent t-conorm is max Now

VO E LE Vx E R x,e ( x ) = f , (xe ( x ) ) Hence, by the law of excluded middle max (xe ( x ) , f , ( X Q ( x ) ) ) = 1 Now i f xe ( x ) = 1 then the result is proven Otherwise i f f , (xe ( x ) ) = 1 then by negation axiom N 4 xe ( x ) =

f , ( f , (xe ( x ) ) ) = f , ( 1 ) = 0 by negation axiom N 1 as required

Elkan [30] somewhat controversially proved a related result for Zadeh's orig- inal min / rnax calculus Elkan's result focuses on the restrictions imposed on membership values if this calculus is to satisfy a particular classical equivalence relating to re-expressions of logical implication Clearly this theorem is weaker than that of Dubois and Prade [20] in that it only concerns one particular choice

of t-norm, t-conorm and negation function

Let f A (a, b) = min(a, b), f v (a, b) = max(a, b) and f , ( a ) = 1 - a For this calculus if YO, cp E LE and Vx E R x,(e,,,,) ( x ) = ~,~(,e,,,,) ( 2 ) then

YO, cp E LE and Vx E R either xe ( x ) = X , ( x ) or X , ( x ) = 1 - X e ( x )

The controversy associated with this theorem stems mainly from Elkan's assertion in [30] that such a result means that previous successful practical ap- plications of fuzzy logic are somehow paradoxical The problem with Elkan's attack on fuzzy logic is that it assumes a priori that vague concepts should satisfy

a specific standard logical equivalence, namely 1 (0 A l c p ) = cp V (70 A l c p )

No justification is given for the preservation of this law in the case of vague concepts, except that it corresponds to a particular representation of logical implication Given such an equivalence and assuming the truth-functional

min / max calculus of Zadeh then the above reduction theorem (theorem 7) follows trivially

In their reply to Elkan, Dubois etal [23] claim that he has confused the notions of epistemic uncertainty and degree of truth Measures of epistemic uncertainty, they concede, should satisfy the standard Boolean equivalences while degrees of truth need not In one sense we agree with this point in that Elkan seems to be confusing modelling the uncertainty associated with the object domain R with modelling the vagueness of the concepts in the underlying description language LE On the other hand we do not agree that you can completely separate these two domains When You make assertions involving

vague concepts then Your intention is to convey information about R (a fact recognized in fuzzy set theory by the linking of fuzzy membership functions with possibility distributions [115]) It does not seem reasonable that questions related to this process should be isolated from those relating to the underlying calculus for combining vague concepts The way in which You conjunctively combine two concepts L1 and L2 must be dependent on the information You want to convey about x when You assert 'x is L1 A La' and the relationship

of this information with that conveyed by the two separate assertions 'x is L1' and 'x is L2' Furthermore, it is not enough to merely state that truth-degrees are different from uncertainty and are (or can be) truth-functional Rather,

we claim that the correct approach is to develop an operational semantics for vague concepts and investigate what calculi emerge Indeed this emphasis on

a semantic approach forms the basis of our main object to Elkan's work The problem of what equivalences must be satisfied by vague concepts should be investigated within the context of a particular semantics It is not helpful to merely select such an equivalence largely arbitrarily and then proceed as if the issue had been resolved

The theme of operation semantics for vague concepts is one that we will return to in a later section of this chapter and throughout this volume However, for the moment we shall take a different perspective on the result of Dubois and Prade (and to a lesser degree that of Elkan) by noting that it provides

an insight into what a strong assumption truth-functionality actually is We also suggest that truth-functionality is a special case of a somewhat weaker assumption formalizing the following property: Functionality assumes that for any sentence 0 E L E there exists some mechanism by which Vx E R

Xe (x) can be determined only from the values {xL (x) : L E LA) (i.e the membership values of x across the basic labels) This notion seems to capture the underlying intuition that the meaning of compound vague concepts are derived only from the meaning of their component concepts while, as we shall see in the sequel, avoiding the problems highlighted by the theorems of Dubois and Prade and of Elkan

DEFINITION 8 The measure v on L E x R is said to be functional ifV0 E

L E there is function fe : [0, lIn -t [O,1] such that Vx E fl ve (x) =

fe ( V L ~ (4, a , VL, ( 4 )

The following example shows that functional measures are not necessarily subject to the triviality result of Dubois and Prade [20]

E X A M P L E 9 Functional but Non-Truth Functional Calculus

Let LA = {L1, La) and for 0 E L E let Ox denote the proposition 'x is 6'

Now let ve (x) denote P (Ox) where P is a probability measure on the set of propositions {Ox : 0 E LE) Suppose then that according to the probability

18 MODELLING AND REASONING WITH VAGUE CONCEPTS

measure P the propositions (L1), and (L2), are independent for all x E R

In this case v is a functional but not truth functional measure For example,

Howevel; since v is defined by a probability measure P then

except when v ~ , (x) = 0 or VL, (x) = 1

Clearly, howevel; ~ L ~ A L ~ (x) can be determined directly from VL, (x) and

V L ~ (x) according to the function fLIALl (a, b) = a Indeed ve (x), for any compound expression 8, can be evaluated recursively from UL, (x) and v ~ , (x)

as a unique linear combinations of vai (x) : i = 1, ,4 For instance,

In General

Hence, we have that

vffi (x) = vffi (x) = ve (x) and

Clearly then v satisfies idempotence and the law of excluded middle, and hence functional calculi are not in general subject to the restrictions of Dubois and Prude's theorem [20]

In [I031 Walley proposes a number of properties that any measure should sat- isfy if it is to provide an effective means of modelling uncertainty in intelligent systems These include the following interpretability requirement:

'the measure should have a clear interpretation that is sufficiently defined to guide assessment, to understand the conclusions of the system and use them as a basis for action, and to support the rules for combining and updating measures'

Thus according to Walley an operational semantics should not only provide

a means of understanding the numerical levels of uncertainty associated with propositions but must also provide some justification for the underlying cal- culus In the case of fuzzy logic [I101 this means than any interpretation of membership functions should be consistent with truth-functionality If this turns out not to be the case then it may be fruitful to investigate new calculi for combining imprecise concepts

In [25] Dubois and Prade suggest three possible semantics for fuzzy logic One of these is based on the measure of similarity between elements and pro- totypes of the concept, while two are probabilistic in nature In this section we shall review all three semantics and discuss their consistency with the truth- functionality assumption of fuzzy logic We will also describe a semantics based on the risk associated with making an assertion involving vague concepts (Giles [34])

A direct link between membership functions and similarity measures has been proposed by a number of authors including Ruspini [89] and Dubois and Prade [25], [28] The basic idea of this semantics is a follows: For any concept 0

it is assumed that there a set of prototypical instances drawn from the universe

R of which there is no doubt that they satisfy 0 Let 'Po denote this set of prototypes for 0 It is also assumed that You have some measure of similarity according to which elements of the domain can be compared Typically this is assumed to be a function S : R2 + [O, 11 satisfying the following properties:

S2 Vx E RS(x,x) = 1 4

The membership function for the concept is then defined to be a subjective measure of the similarity between an element x and the closest prototypical element from 'Po:

Clearly then if x E Po then xo (x) = 1 and hence if # 8 then

sup {xe (x) : x E R) = 1 Also, if % = 0 then Vx E R xo (x) = 0 and hence according to prototype semantics all non-contradictory concepts have normalised membership functions

We now consider the type of calculus for membership functions that could

be consistent with prototype semantics Clearly this can be reduced to the problem of deciding what relationships hold between the prototypes of concepts generated as combinations of more fundamental concepts and the prototypes

of the component concepts In other words, what are the relationships between and PQ, between 'Per\, and 'Po and 'P,, and between 'Pev, and 'Po and P,

20 MODELLING AND REASONING WITH VAGUE CONCEPTS

For the case of 18 it would seem uncontroversial to assume that PYe C (%)" Clearly, a prototypical not tall person cannot also be a prototypical tall person

In general, however, it would not seem intuitive to assume that PYe = (Po)"

since, for example, someone who is not prototypically tall may not necessarily

be prototypically not tall

For conjunctions of concepts one might naively assume that the prototypes for 8 A p might correspond to the intersection Pe n 'P, In this case it can easily

be seen that:

However, on reflection we might wonder whether a typical tall and medium

person would be either prototypically tall or prototypically medium This is essentially the basis of the objection to prototype theory (as based on Zadeh's min-max calculus) raised by Osherson etal [75] For example, they note that when considering the concepts pet and fish then a guppie is much more prototypical of the conjunction pet jsh than it is of either of the conjuncts Interestingly when viewed at the membership function level this suggests that the conjunctive combination of membership functions should not be monotonic (as it is for t-norms) since we would intuitively expect guppie to have a higher membership in the extension of petjsh than in either of the extensions of pet orjsh

In the case of disjunctions of concepts it does seem rather more intuitive that POv, = Pe U P, For example, the prototypical happy or sad people might reasonably be thought to be composed of the prototypically happy people together with the prototypically sad people In this case we obtain the strict equality:

Osherson etal [75] argue against the use of prototypes to model disjunctions using a counter example based on the concepts wealth, liquidity and investment

The argument presented in [75] assumes that wealth corresponds to liquidity

or investment, however, while there is certainly a relationship between these concepts it is not at all clear that it is a disjunctive one

E X A M P L E 10 Suppose the universe $2 is composed ofjvepeople:

with the following similarity measure S